The 2 Aspects

There’s 2 Aspects to things in general. I will call them Mapping Out and Mapping In, in titlecase so you know they’re distinct concepts.

warmup: 0 -> 1

Here, 0 is an initial object and 1 is a terminal object. 0 is Mapped Out of because it’s 0 -> and not -> 0. 1 is Mapped Into because it’s -> 1 and not 1 ->. The defining property of an initial object is that it has 1 map to every other object. The defining property of a terminal object (1) is that it has 1 map from every other object.

Broken Symmetries

• Injective functions have inverses. But the dual statement, that surjective functions always have (at least 1) inverse, is equivalent to the Axiom of Choice!
• a AND (NOT a) = false is just the law of noncontradiction. A simple theorem (unless you’re Graham Priest), but the dual statement a OR (NOT a) = true is the Law of Excluded Middle, which can be derived from a form of the Axiom of Choice, and in particular is not constructively valid.
• Freyd’s Adjoint Functor Theorem sometimes fails to have a left adjoint (Mapping Out), but for foundations-of-mathematics-y reasons: If the left adjoint existed, it would be too big to be a set. No such difficulty for right.
• Googling epimorphism too large to form set gives a lot of stuff. Nothing similar for monomorphism too large to form set
• infinitesimals are all near the standard number 0, but unlimited numbers have no standard part.
• related to one before, there’s that 0 is the smallest, but there’s no largest.
• positive and negative are simple flips additively, but negative times negative is positive.

The following is from Paul Garrett, Section 2:

When we look at colimits and coproducts here, it is important to see that, while at an abstract level these things are just the arrows-reversed versions of limits and products, for many classes of naturally-occurring objects there is a sharp asymmetry. For example, while limits are subobjects of products, colimits are quotients of coproducts. In many situations, quotients are more abstract entities than are subobjects. This can be explained from a set-theoretic viewpoint, since elements of a subset are the same sort of thing as elements of the original set, since they are elements of the original set, while elements of quotients are sets of elements of the original.

In particular, in many cases colimits are fragile, and need further details or hypotheses to give us helpful outcomes. For example, while all subspaces of Hausdorff topological spaces are Hausdorff, quotients of Hausdorff topological spaces need not be Hausdorff.

submodules of finitely-generated free modules over principal ideal domains are still free, while quotients certainly need not be.

Indeed, the smooth general use of products and limits is not matched by any similar smoothness in treatment of the arrow-reversed coproducts and colimits, below.